Barbara Thompson's 1995 AGU Spring Meeting Poster

Numerical simulation of Alfven waves

Wave integration - fluid motion

The wave equations for an inertial Alfven wave take the form of Maxwell's equations in a dielectric medium. A combination of Fourier transforms and numerical integration can produce solutions to the system. Lysak [1991] calculated the resonant frequencies of the system by Fourier transforming in time and in both perpendicular directions and integrating along the field lines. These results represented single resonant wave modes.

The response to time-dependent perturbations of the magnetic field can be modeled using an algorithm which integrates time and along the geomagnetic field. Assuming the background plasma density is fairly homogeneous in the direction perpendicular to the field lines (over length scales comparable to the perpendicular wavelength), a Fourier transform can reduce behavior in the perpendicular direction to a series of perpendicular wave numbers. The leapfrog scheme is shown in Figure 9 of the poster - since the first-order derivative of a function in time is represented (on average) at the center of the cell, it is more accurate to stagger the cells, so that the derivatives in vector potential are represented at grid points of the scalar potential, and vice versa. Using the staggered grid, the method is second-order accurate and has zero amplitude error.

If the system cannot be transformed in the perpendicular direction, derivatives in the third variable, y, form a tridiagonal system. Time and parallel distance are still integrated with the leapfrog scheme, while the tridiagonal system is solved simultaneously.

Electron integration - particle motion

Particle electrons are inserted into the system, and are accelerated by the parallel electric field (the local gyrofrequency of electrons is much larger than the wave frequency, so the perpendicular fields are not a factor; in some regions it may be important for protons). The magnetic field exerts a mirror force, but represents no net gain in energy, since the vXB force acts perpendicular to the velocity and does no work.

The electrons are tracked using dynamical arrays called linked lists. This method of "pushing" and "popping" electrons in and out of the system represents local acceleration.

Conserving energy - numerical scheme

At the end of each time step, after the wave equations have been iterated and each electron on the linked list has been accelerated, the energy gained by each electron is deducted from the local wave energy density. The total energy in a grid cell found using the energy density - conservation of energy implies that this amount much decrease by the amount of energy gained by an electron.

While this has the effect of conserving energy, it is not completely self-consistent. Although the waves affect the particles through acceleration and the particles in turn affect the wave by deducting the change in energy, a full consideration of Maxwell's equations using source terms is necessary.

Return to Summary Page

Other Sections

Alfven waves and auroral systems - simple overview

System properties and physical behavior

Deriving the wave equations including electron inertial effects

Dielectric Treatment of Alfven waves

Electron effects and commonly observed distributions

Chaotic behavior of system

References - please see poster